# condition number help?

• February 9th, 2013, 04:12 PM
gspease839
condition number help?
im instructed to calculate the condition number. according to my slides is is calculated by ||A|| ||A^-1|| = condition number. what exactly is this? absolute matrix A times absolute inverse matrix A??
• February 9th, 2013, 05:01 PM
Norm
Re: condition number help?
Can't see where your java programming problem is here, so I moved it to the algorithms section.
• February 9th, 2013, 05:43 PM
gspease839
help with matrix condition number
im instructed to calculate the condition number. according to my slides is is calculated by ||A|| ||A^-1|| = condition number. what exactly is this? absolute matrix A times absolute inverse matrix A??
• February 9th, 2013, 05:52 PM
helloworld922
Re: condition number help?
That notation looks familiar, but I can't recall what it is off the top of my head. Try asking what it is over at Math Stack Exchange.
• February 9th, 2013, 06:40 PM
Zaphod_b
Re: help with matrix condition number
Quote:

Originally Posted by gspease839
im instructed to calculate the condition number. according to my slides is is calculated by ||A|| ||A^-1|| = condition number. what exactly is this? absolute matrix A times absolute inverse matrix A??

The ||.|| notation means the matrix norm. You can look it up in your favorite text on linear algebra. (Or you can poke around on the web and see if you can find a description that suits your sensibilities.)

I'll dumb-down the language from strict linear algebraic terminology:

If the matrix is diagonalizable, it turns out that the condition number is equal to (and is "usually" calculated by) the absolute value of the eigenvalue with largest absolute value divided by the absolute value of the eigenvalue with smallest absolute value.

If the matrix is not diagonalizable, it turns out that the condition number is equal to (and is "usually" calculated by) the absolute value of the singular value with largest absolute value divided by the absolute value of the singular value with smallest absolute value.

The significance of the condition number of a matrix in many applications goes something like this:

If the condition number of an invertible matrix is "close to" 1 (the eigenvalues are all of similar order of magnitude) the numerically calculated inverse of the matrix isn't terribly sensitive to small errors in the input matrix coefficients or to roundoff error in the calculations.

One result is that solutions of systems of equations whose coefficients are the elements of matrix with this property (eigenvalues all of similar order of magnitude) can be calculated without being terribly sensitive to small errors in the input coefficients or to roundoff errors in the calculations. Such a matrix is said to be "well conditioned."

On the other hand, if an invertible matrix has a high condition number ("lots greater than 1"), the matrix is said to be "ill-conditioned," and calculation of the inverse matrix is extremely sensitive to small input errors or roundoff errors in the calculations. Calculated solutions of systems of equations whose coefficients are elements of an ill-conditioned matrix are extremely sensitive to small input errors or roundoff errors in the calculations.

Cheers!

Z